1. Let

where a > 0, be an indefinite quadratic form, so that d = b2 − 4ac > 0. A classical theorem of Minkowski states that, if (x0, y0) is any pair of real numbers, there are numbers (x, y) congruent (mod 1) to (x0, y0), such that

and, more recently, Davenport has shown that this theorem can be sharpened for certain special f, for instance that it is always possible to satisfy

In a previous joint paper (‘The dissection of rectangles into squares’, by R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte, Duke Math. J. 7 (1940), 312–40), hereafter referred to as (A) for brevity, it was shown that it is possible to dissect a square into smaller unequal squares in an infinite number of ways. The basis of the theory was the association with any rectangle or square dissected into squares of an electrical network obeying Kirchhoff's laws. The present paper is concerned with the similar problem of dissecting a figure into equilateral triangles. We make use of an analogue of the electrical network in which the ‘currents’ obey laws similar to but not identical with those of Kirchhoff. As a generalization of topological duality in the sphere we find that these networks occur in triplets of ‘trial networks’ N1, N2, N3. We find that it is impossible to dissect a triangle into unequal equilateral triangles but that a dissection is possible into triangles and rhombuses so that no two of these figures have equal sides. Most of the theorems of paper (A) are special cases of those proved below.

1. In a recent paper (1) the reflexion of surface waves by a submerged plane barrier has been considered, and it has been shown that there is a reflected wave of finite amplitude at a great distance from the barrier, so that the coefficient of reflexion (the ratio of the amplitudes, at a great distance from the barrier, of the reflected and incident waves) is finite. In this paper it is shown that in the case of reflexion by a submerged circular cylinder, the coefficient of reflexion is zero, with the result that the only effect of the obstacle at a great distance is that there is a phase-difference between the incident and transmitted waves, their amplitudes being the same. Only one case has been worked out in detail numerically, and the method employed is such that the condition that the section of the cylinder must be a stream-line has been satisfied only to an approximation; it is, however, shown that the coefficient of reflexion is zero in the general case.

1. The problem of determining the distribution of stress in a semi-infinite solid medium when its plane boundary is deformed by the pressure against it of a perfectly rigid cone is of considerable importance in various branches of applied mechanics. It arises in soil mechanics where the cone is the base of a conical-headed cylindrical pillar and the semi-infinite medium is the soil upon which it rests (1). In this instance the distribution of stress in the soil is known to be more or less similar to that calculated on the assumption that the soil is a perfectly elastic, isotropic and homogeneous medium, at least if the factor of safety of a mass of soil with respect to failure by plastic flow exceeds a value of about three (2). The same problem occurs in the theory of indentation tests in which a ductile material is indected by cylindrical punches with conical heads (3).

It is shown that under certain conditions convective flow may occur in fluid which permeates a porous stratum and is subject to a vertical temperature gradient, on the assumption that the flow obeys Darcy's law. The criterion for marginal stability is obtained for three sets of boundary conditions, and the motion described. If such convection occurs in a stratum through which a bore-hole passes, the usual method of calculation of the heat flow must be modified, but in general the correction will not be large.

The conditions for equilibrium in an elastically stressed hexagonal aeolotropic medium (transversely isotropic) are formulated, and solutions are found in terms of two ‘harmonic’ functions ø1, ø2, which are solutions of

ν1, ν2 being the roots of a certain quadratic equation.

It is also shown that in the case of axially symmetrical stress systems the solution may be expressed in terms of the third-order differential coefficients of a single stress function Φ.

The solutions for an isotropic medium may be deduced as a special case.

The problems of nuclei of strain in such a hexagonal solid are solved, and the results for zinc and magnesium contrasted with those for an isotropic solid.

Recent studies by Debye (1), Bloch (2), Stoner (3) and Guggenheim (4) expressing somewhat divergent views suggest that there is still some confusion and uncertainty in the method of application of the thermodynamic argument to magnetic phenomena. The first effective contribution to the subject was made by Larmor (5) over fifty years ago, and developing this further he finally proposed in 1929 a sufficient basis for a general theory of the whole range of phenomena. At about the same time (1927), and apparently in ignorance of Larmor's earlier work, Debye(1) also gave a brief outline of the thermodynamic aspects of the subject, starting, however, from much more tentative magnetic ideas and formulae which are now known to be without adequate physical justification. In 1933, Bloch (2) elaborated the argument still further, using, however, basic ideas more akin to those employed by Larmor, but derived by a very specialized argument. Then in 1935, using the same fundamental equation as Debye, Stoner revived the discussion with a systematic survey of the theoretical results bearing on the thermodynamic aspects of the subject, attempting later (6) to justify the basic magnetic ideas by a discussion which he now admits is not entirely satisfactory.

Heisenberg and Pauli (1) have shown how to quantize field theories derived from a Lagrangian containing first derivatives of the field quantities only. The present paper extends the theory of quantization of fields to the case of higher order Lagrangians, i.e. Lagrangians in which higher derivatives than the first appear. It is shown how such field equations can be put into Hamiltonian form and how the quantization can subsequently be carried out. Both the cases of Einstein-Bose and Fermi-Dirac quantization are discussed. It is established that the quantization is relativistically invariant and consistent with the field equations. An interesting feature of the present theory is that the Hamiltonian proves to be different, in general, from the integral of the 4–4 component of the energy momentum tensor.

Some new measurements of isotropic turbulence produced behind a biplane grid have been made at high Reynolds numbers, and these results are compared with the predictions of the theory of local isotropy developed by A. N. Kolmogoroff. The transverse double-velocity correlation has been measured at mesh Reynolds numbers up to 3·2 × 105, and the observed form agrees well with the predicted form. Measurements of the skewness factor of velocity differences over finite intervals have also been made, and the factor is nearly constant and equal to −0·38, if the interval is small compared with the integral scale. The invariance of dimensionless functions of the velocity derivatives has been confirmed for the flattening factor of ∂u/∂x, namely,

which is nearly constant over a wide range of conditions. It is concluded that the theory of local isotropy is substantially correct for isotropic turbulence of high Reynolds number.

When impact occurs between clean metal surfaces, plastic flow of the metal usually takes place at the points of real contact, so that the pressures developed are as high as the dynamic yield pressure of the metal concerned. Early experiment show that if the surfaces are covered with a thin film of a highly viscous liquid, the pressures developed and transmitted through the liquid film may be sufficiently great to produce plastic deformation of the metal, even though no metallic contact occurs through the film (1). The existence of these high pressures in the liquid layer means that extremely high rates of flow and shear may be developed in the liquid film, and that the energy dissipated in overcoming viscous flow may lead to an appreciable temperature rise in the liquid. Even in much gentler impacts, where plastic deformation of the metal surfaces does not occur, very high pressures, rates of flow and shear, etc., may be developed in the liquid film. These effects are of great interest in any study of collisions through liquids; they are of particular significance in the study of the mechanism of detonation of liquid explosives by impact.

Several methods have been developed in recent years in the Cavendish Laboratory for the absolute measurement of fast neutron flux. All depend on observing the effects of elastic collisions between neutrons and light nuclei. The methods fall into two categories according as individual recoil nuclei are counted (1, 2), or the total ionization current due to the recoils is measured (3). In order completely to interpret the experimental results, it is necessary to know the cross-section for scattering and the angular distribution of the recoil nuclei. The total number of recoil nuclei and their energy distribution are then determined for a known incident neutron spectrum. For precise neutron flux measurements, recoil protons are invariably studied, as the neutron-proton scattering cross-section has been measured over a wide range of energies (4, 5), and the angular distribution of the recoils is isotropic in the centre of gravity space for neutrons of energy less than about 10 MeV. (6, 7). This makes the reduction of the experimental results particularly simple and certain.

It is shown how the configurational partition functions of certain assemblies can be rigorously constructed. From them it is proved that the terms in the expression for the free energy of the assembly depending on the surface enclosing the assembly are of a smaller, order than N, N being the total number of atoms. This justifies the usual ignoration of surface effects in studying the bulk properties of a large assembly of atoms.